protection for sale to oligopolists - kuweb.econ.ku.dk/epru/protection4sale...

Protection for Sale to Oligopolists�

Elena Paltsevay

Preliminary and incomplete.

Abstract

This paper modi�es Grossman and Helpman�s canonical "Protection for Sale" modelby allowing demand linkages and oligopolistic competition. It shows that increased sub-stitutability between products weakens interest groups�incentives to lobby. For the caseof one organized and one unorganized industry, we obtain a particularly simple result:as product substitutability increases, the protection of the organized industry�s productfalls, whereas the protection of the unorganized sector�s product increases.Empirical studies of the "Protection for Sale" model have suggested that the U.S.

government�s trade policy decisions are overwhelmingly determined by a concern forwelfare maximization; the alternative interpretation of the paper is that the originalmodel overstates the lobby groups�desire for protection.

�I am grateful to Tore Ellingsen and Victor Polterovich for advice and encouragement. I also thankAvinash Dixit, Gene Grossman, Giovanni Maggi and Torsten Persson for valuable comments, as well asseminar participants at Stockholm School of Economics and Princeton University. Jan Wallander�s and TomHedelius�Research Foundation is gratefully acknowledged for �nancial support. All remaining errors are myown.

yAddress: Department of Economics, University of Copenhagen, Studiestrade 6, DK-1455 Copenhagen,Denmark, [email protected]

Protection for Sale to Oligopolists 2

1 Introduction

Recent theories of endogenous trade policy suggest that trade policy is determined throughinteraction between organized lobby groups and privately interested policy-makers, ratherthan by a benevolent social-welfare maximizer. While there are many variations on thetheme (see Rodrik (1995) for a review), Grossman and Helpman�s (1994) "Protection forSale" model has been the most in�uential by far. Grossman and Helpman (henceforth GH)explicitly describe a mechanism through which interest groups�contributions in�uence thepolicy-maker�s decision for trade protection, providing micro-foundations for the previousapproaches. The model�s predictions for the equilibrium protection pattern relate the indus-try�s trade tari¤ to import demand elasticity, state of organization, import penetration andother variables, thereby providing a coherent framework for empirical testing.

As is well known, GH neglect some important issues. In particular, they abstract fromproduction linkages and strategic market interactions. Indeed, factor-speci�c production inGH implies that di¤erent lobbies do not compete in the factor market. Hence, their interestsare only opposed to the extent that each industry wants to increase its pro�ts by raising theprice of its own good, while all other organized industries aim at reducing the same price inorder to increase their members�consumer surplus. That is, political competition among thelobbies arises purely from the desire of the members of di¤erent industry groups to protecttheir interests as ordinary consumers. This feature of GH has caused concerns about theinterpretation of the model.1 To add a more realistic justi�cation for the political rivalryamong the organized groups, Gawande and Bandyopadhyay (2000) introduce supply-sideinteractions through a single importable intermediate input. Cadot, de Melo and Ollarreaga(2001) strengthen the degree of inter-industry interactions even further, assuming that theindustries compete for a common scare resource and each good is produced using othergoods as intermediate inputs. These extensions help obtain predictions consistent with theobserved stylized facts, e.g., escalation of protection rates with the degree of processing, orhigher average protection for poor countries.

This paper instead focuses on demand-side interactions, addressing the rivalry betweenthe organized groups that arises due to the substitutability between goods. It studies theimpact of demand linkages on the determination of trade policy, and the intensity of inter-industry lobbying competition.

The consumers�utility function in the original GH model is assumed to be quasi-linear,implying that the demand for each product is independent of the prices of the other products.Besides, the GH model considers a small and open economy, so that the producers faceperfectly elastic demand. Thus, the prices in other industries do not a¤ect the producers�incentives to lobby. To address the inter-industry competition in lobbying and its impacton the trade policy, resulting from the demand-side interactions, I relax these assumptionsand allow for both demand linkages and imperfectly competitive industries. More precisely,I consider a utility function that admits cross-price e¤ects on demand and assume thateach good is produced by an international oligopoly and sold in internationally segmentedmarkets.

The �rst part of the paper shows that the presence of substitutes may reduce interest

1See e.g. Baldwin and Robert-Nicoud (2006).

1. Introduction 3

group incentives to lobby due to competition in the goods market. If demands are interde-pendent, an increase in the price of a good causes demand to shift towards its substitutes.A decrease in the price of the substitute has a similar e¤ect. The interest group takes thisshift into account when lobbying the government to increase the price of its own good (asa producer receiving pro�t from selling this good) and reduce the price of all other goods(as a consumer who wants to maximize her utility). Therefore, the lobbying strategy of anorganized industry becomes less aggressive. To put it di¤erently, introducing substitutabilityproduces more aligned interests between the interest groups. As a result, other things equal,economies producing closer substitutes (or having a more competitive industry structure)should experience more moderate protection rates. That is, with an increase in the degreeof substitutability, the protection of the organized industries falls, and the protection of thenon-organized industries increases, relative to the �rst-best benchmark.

The result suggests a new explanation for an observed empirical puzzle. Studies devotedto empirical testing of the GH theory �nd that the government puts very low weight onthe campaign contributions relative to the welfare loss. That is, the government behavesalmost as a social welfare maximizer and trade policy deviates surprisingly little from the�rst-best,2 thus causing a concern about the empirical signi�cance of the GH model. E.g.,Gawande and Krishna (2002) write that "..it is enough to cast doubt on the value of viewingtrade policy determination through this political economy lens". There have been severalattempts at modifying the model to explain low protection rates. For example, Gawandeand Bandyopadhyay (2000) introduce political competition between the upstream and down-stream producers, and Gawande, Krishna and Robbins (2004) introduce counter-lobbying bythe foreign organized groups. This paper provides an alternative explanation, suggesting thatsmaller deviations from the �rst-best protection rates may result from the weaker incentivesto lobby caused by the substitutability e¤ects.

The second part of the paper addresses the impact of product substitutability on lobbyformation. The original GH model assumes an exogenous lobby group structure. A naturalquestion to ask is thus why are some industries organized and others not. In the GH model,the interests of di¤erent industry groups are opposed to each other, so if an additionallobby formation stage is introduced in their model, all industries would get organized (in theabsence of the lobby formation costs).3

However, in the presence of demand linkages, at a su¢ ciently high degree of substitutiona non-organized industry becomes protected without paying for it. The reason is that theorganized industry cannot lobby to increase its own price without losing a substantial partof its consumers switching to the cheaper (but similar) good. For the same reason, it cannotlobby for dropping the price of the substitute. Thus, the non-lobbying industry gets a freeride on the lobbying industry e¤orts. If instead both these substitute-producing industriesare organized, they both contribute to the government for (potentially higher) protection.Comparing these two outcomes, it turns out that the industry may better o¤ not beingorganized. As a result, with endogenous lobby formation, fewer industries get organized andlobbying becomes less intense. We �nd this free-riding e¤ect to be present for industrieswith relatively dispersed ownership, which is in line with Olson�s (1965) interest group size

2E.g. Goldberg and Maggi (1999); Gawande and Bandyopadhyay (2000).3Like GH, we neglect any intra-industry organizational con�ict.


hypothesis.This paper is not the �rst to endogenize lobby formation in the GH model or discuss the

possibility of free-riding. Mitra (1999) adds an initial stage to the GH model, letting theowners of each speci�c factor decide in Nash equilibrium whether it is pro�table to incur a�xed cost of forming a lobby. Magee (2002) employs a two-stage game, where in the �rststage, industry representatives and the policy maker determine the tari¤ schedule and, in thesecond stage, every �rm in the industry decides whether to contribute to the lobbying e¤ort(detecting is in�nitely punished). Both these papers address the collective action problem atthe intra-industry level. We, instead, relate lobby formation to the inter-industry demandlinks, thereby avoiding the issue of exogenous �xed costs.

The remainder of the paper is organized as follows: Section 2 describes the model setup,Section 3 discusses the equilibrium structure of protection, Section 4 analyzes the e¤ect ofthe degree of substitution on the extent of protection, Section 5 studies the question of thelobby formation in the presence of substitute goods and Section 6 concludes.

2 The model

There are m+1 goods being produced in an open economy. The domestic price of good i isdenoted pi. Good 0 is taken to be a numeraire, so p0 = 1. The individuals populating theeconomy have identical preferences represented by the quasi-linear utility function

U(x0; x1; :::; xm) = x0 + U(x1; :::; xm); (1)

where xi denotes consumption of good i: The original GH model uses the separable utilityfunction

U(x0; x1; :::; xn) = x0 +nXi=1

ui(x1; :::; xn);

which implies that demand functions are independent of the prices of the other goods. I relaxthis assumption allowing for the cross-price e¤ects. More precisely, I adopt the quadraticsub-utility function:

U(x1; :::; xm) =mXk=1

xk � 1=2mXk=1

x2k � �mXk=1

mXj=k+1

xkxj ;

where � 2 [0; 1] re�ects the substitutability between goods x1; :::xm. As is well known, theassociated inverse demand function for each non-numeraire good i is

pi = 1� xi � �mXj 6=ixj :

For � 2 (0; 1), considering only interior solutions, the resulting domestic demand functionfor good i is

di(p1; :::; pm) =

(1� �)� ((m� 2)� + 1) pk + �mPj 6=kpj

((m� 1)� + 1) (1� �) :

2. The model 5

In words, domestic demand is linear in prices of non-numeraire goods, decreasing in the priceof the own good and increasing in the prices of the other goods. That is, good i and goods1; 2::i� 1; i+ 1; ::m are imperfect substitutes.

Due to the quasi-linearity of the utility function, any income e¤ect is totally captured bythe consumption of good 0: That is, when optimally spending an amount E, each individualobtains demand functions

xi = di(p); i 2 f1; :::;mg

x0 = E �mXk=1

pkdk(p);

where p is the vector of domestic prices (p1; :::; pm): The associated indirect utility functionis

V (p) = E �mXk=1

pkdk(p) + U(d1(p); :::; dm(p)):

The numeraire good is produced using labor only, with an input-output coe¢ cient of 1.As the non-numeraire good is freely traded in a perfectly competitive international market,the wage rate in this economy is equal to 1. The other m goods are produced by a CRStechnology using labor alone, but are sold at internationally segmented oligopolistic markets.More precisely, good k is supplied by nk > 0 identical domestic �rms and n�k > 0 identicalinternational �rms, competing in quantities. Thus, the total number of �rms operating insector k is given by

Nk = nk + n�k:

It takes ck units of labor to produce one unit of good k; both at home and abroad.4 Weassume that each �rm is a pure pro�t maximizer and that the number of �rms is �xed, sothat no entry or exit decisions are taken.

Each individual is endowed with some labor and may also own some claims to the pro�tof a �rm in at most one industry. These claims are indivisible and non-tradable.

The government introduces trade taxes and/or subsidies and redistributes the result-ing net revenue from all these taxes equally among the voting population. For each non-numeraire good, the domestic and the international market are segmented and productionis characterized by the constant marginal costs, ck. As a result, �rms�production decisionsare made separately for the domestic and the international market. The trade tax revenueis separable in the import and export tax and, as a result, so is consumer welfare. Thus,the government can set import and export trade policies separately. In this paper, we con-centrate on the government decision about import tari¤s, which determines the productionand consumption in the domestic market. We also assume that the foreign government doesnot impose any export tari¤s on its �rm, so that in the absence of trade intervention of thedomestic government, �rms at home and abroad are equally e¢ cient. As we do not studythe strategic interaction between the home government and foreign governments in settingtrade policy, this assumption does not a¤ect the model�s qualitative results.

4The assumption that the domestic and the foreign �rm have the same unit cost ck is made for computa-tional convenience. It does not change the predictions of the model.


We denote the trade tari¤ in sector k by �k: Then, in Cournot-Nash equilibrium, each ofthe nk domestic �rms in sector k solves the pro�t maximization problem

maxqik

�ik = qik

241� nkXj=1

qjk �n�kXj=1

qj�k � �Xs 6=k

0@ nsXj=1

qjs +

n�sXj=1

qj�s

1A35� ckqik; (2)

where qik denotes the production of domestic �rm i in sector k, and qj�k denotes the production

of foreign �rm i in sector k: Similarly, each of the n�k foreign �rms in sector k solves theproblem

maxqi�k

�i�k = qi�k

241� nkXj=1

qjk �n�kXj=1

qj�k � �Xs 6=k

0@ nsXj=1

qjs +

n�sXj=1

qj�s

1A35� (ck + �k) qj�k : (3)

Solving the system of these equations for all sectors yields the equilibrium quantities producedby each �rm. The symmetry assumption implies that �rms within each sector choose thesame equilibrium output at home,

qik(� ) = qjk(� ) � qk(� );

and abroadqi�k (� ) = q

j�k (� ) � q

�k(� );

where � denotes the vector of the trade tari¤s (�1; :::�m). Similarly, in industry k, each �rmin the same country earns the same pro�ts, �k(� ) for a domestic �rm and ��k(� ) for a for-eign �rm, respectively. Moreover, from the �rst-order conditions of the pro�t maximizationproblems for the domestic �rm (2) and the foreign �rm (3) in sector k, it follows that thepro�t is given by

�k(� ) = q2k; (4)

��k(� ) = (q�k)2 :

Note also that the market clearing condition implies that

dk(p(� )) = nkqk(� ) + n�kq�k(� ):

The amount of the tax revenue collected by the government (and redistributed to thecitizens) is equal to

r(� ) =mXk=1

�kn�kq�k(� ): (5)

Thus, individual income is the sum of wages, government transfers and possibly claims to adomestic �rm�s pro�t.

The owners of �rms in the same industry may choose to organize and form a lobby grouptrying to in�uence the government in its decision about trade policies. The joint welfare ofthe members of such a group i comprising share �i of the total population is

Wi(� ) = li + ni�i(� ) + �i

"r(� )�

mXk=1

pkdk(p) + U(d1(p(� )); :::; dm(p(� )))

#; (6)

3. The structure of protection 7

where li denotes the total labor endowment of group i members. Note that the vector ofequilibrium prices is, in turn, a function of trade tari¤s.

Each lobby i may contribute to the government an amount Ci(� ) conditional on the tradepolicy vector, or, equivalently, the domestic price vector.

The objective function of the government is

G (� ) =Xi2L

Ci(� )+aW (� );

where L is the exogenously given set of organized sectors, a > 0 is the weight the governmentattaches to aggregate welfare in the economy, and

W (� ) = l +mXk=1

nk�k(� ) +

"r(� )�

mXk=1

pkdk(p(� )) + U(d1(p(� )); :::; dm(p(� )))

#(7)

is aggregate welfare in the economy.Trade policy is determined in a two-stage game. In the �rst stage, lobbies simultaneously

announce their contribution schedules, i.e., the amount contributed as a function of theimport tari¤s vector (note that each strategy of a lobby is a function on the price simplex).In the second stage, the government chooses policy by maximizing its objective function overthe suggested contribution schedules.

The resulting equilibrium is a set of contribution functions, each maximizing the welfareof the respective lobby members given the schedules of all other lobbies and the anticipatedtari¤ decision of the government and the price vector maximizing the government�s objectivefunction under these contribution schedules.

3 The structure of protection

The problem has the structure of a menu-auction, as characterized by Bernheim and Whin-ston (1986). Hence, if contribution schedules are di¤erentiable, they are locally truthfularound equilibrium, i.e., a marginal change in each lobby�s contribution to the governmentresulting from a small policy change is exactly equal to the respective marginal change ofthis lobby�s welfare.

Local truthfulness combined with the optimality of the equilibrium price vector for thegovernment results in the condition (equation (12) in GH)X

i2LrWi(� ) + arW (� ) = 0: (8)

Substituting the tari¤ revenue (5) and the expression for lobby i�s welfare (6) into (8)yields

(a+ �L)r"mXk=1

�kn�kq�k(� )�

mXk=1

pkdk(p(� )) + U(d1(p(� )); ::; dm(p(� )))

#(9)

+

mXk=1

(Ik + a)nkr�k(� ) = 0;


where Ik is an indicator function taking the value of 1 if industry k is organized, and 0otherwise, and �L =

Pi2L �i is the total share of population in the organized industries.

Simplifying the system (9) (the detailed derivation can be found in the Appendix) entailsthe following form for each equation j in (9):

�mXk=1

�k@mk(� )

@� j=

mXk=1

(Ik + a)

(a+ �L)nk@�k(� )

@� j�

mXk=1

@pk@� j

dk(p(� )) + n�jq�j (� ); (10)

wheremk(� ) = n

�kq�k(� )

denotes total imports in sector k: In what follows, we shall study the properties of the systemof equations (10).

Before proceeding, we need to establish several intermediate results concerning the re-sponsiveness of domestic and foreign supply to the trade tari¤s.

Lemma 1 The equilibrium quantity produced by the foreign �rm in sector k decreases inthe trade tari¤ on its own good, �k; and weakly increases in the trade tari¤ on the substitutegoods, ��k. That is, @q�k(� )=@� j < 0 if i = k and @q

�k(� )=@� j � 0 if i 6= k:

The equilibrium quantity produced by the domestic �rm in sector k increases in the tradetari¤ on each good i = 1; :::;m: That is, @qk(� )=@� j > 0 if i = k and @qk(� )=@� j � 0 ifi 6= k:

Proof. See the Appendix.Lemma 1 follows from two observations. First, consider the own-tari¤ e¤ect. Due to

Cournot competition and linear demand functions, the home and foreign outputs in sectork are strategic substitutes. A domestic import tari¤ in sector k shifts the foreign �rm�sresponse curve in the south-west direction. Therefore, an increase in �k reduces the outputof the foreign �rm and increases the output of the domestic �rm in sector k. Now, turn tothe cross-tari¤ e¤ect. The output produced by either a domestic or a foreign �rm in sector kand the output produced by a foreign �rm in sector j are strategic substitutes. As a result,an increase in � j induces higher output of both domestic and foreign �rms in sector k.

As the domestic pro�ts are given by relation (4), we immediately obtain the followingcorollary.

Corollary 2 The equilibrium pro�t of the domestic �rm in sector k increases in the tradetari¤ on each good i = 1; :::;m. That is, @�k(� )=@� j > 0 for all k; j:

Let us now turn to the analysis of equation (10). First, consider the case of a separableutility function, so that � = 0.5 This immediately implies that @mk(� )=@� j = @�k(� )=@� j =

@pk=@� j = 0 for k 6= j. In this case, equation (10) becomes

� j =1

(�@mj(� )=@� j)

�(Ij + a)

(a+ �L)nj@�j(� )

@� j��@pj@� j

dj(p(� ))� n�jq�j (� )��; 6 (11)

5 It parallels the original GH setting in an imperfectly competitive environment.6 It is equivalent to

�kpk=(2Ik + 2a)

(a+ �L)

Xk(� )

mk(� )

1�� @mk(�)@�k

=mk(�)pk

�� + 1

pk@mk(�)@�k

�@pk@�k

dk(p(� ))� n�kq�k(� )�;

3. The structure of protection 9

and in the absence of lobbying, the �rst-best trade tari¤s are determined by

�0j =1

(�@mj(� )=@� j)

�nj@�j(� )

@� j��@pj@� j

dj(p(� ))� n�jq�j (� )��: (12)

Due to imperfect competition in the goods markets, free trade is no longer socially optimal.But, as in the original GH model, the organized industries experience higher protectionthan in the �rst-best equilibrium. In turn, the non-organized industries are underprotected.Indeed, other things equal, equations (11) and (12) di¤er by the term

1

(�@mj(� )=@� j)nj@�j(� )

@� j; (13)

having the factor (1 + a)=(a + �L) in the latter equation. From Lemma 1, it follows that�@mj(�)=@� j > 0; @�j(�)=@� j > 0 and thus, the entire term (13) is positive. As longas the organized industries do not comprise the entire population (�L < 1), the factor(Ij + a) = (a+ �L) exceeds 1 for an organized industry (Ij = 1) but falls below 1 for anunorganized industry (Ij = 0). Therefore, other things equal, for an organized industry thetari¤ is higher than the �rst-best tari¤. Similarly, for a non-organized industry the tari¤ setin the presence of lobby groups is lower than the �rst-best tari¤. Note that if all industriesare organized and every voter belongs to some lobby (�L = Ij = 1 8j), the protection ratesare equal to the �rst-best ones. That is, like in GH, the lobbying e¤orts of di¤erent industriesexactly o¤set each other.

Now, let us see how the relaxation of utility separability in�uences the equilibrium tari¤s.Once more, we start by establishing an auxiliary result �we want to characterize the matrixof the derivatives of the import with respect to the trade tari¤s. Denote it by M

M = (Mkj) =

�@mk

@� j

�; 1 � k; j � N:

Lemma 3 The matrix M is invertible and its inverse M�1 is non-positive, i.e., all entriesof M�1are non-positive.

Proof. See the Appendix.We use Lemma 3 to obtain the expression for the trade tari¤s from the system (10).

System (10) in matrix form becomes

�M� = ��@mk

@� j

�(� j) (14)

=

mXk=1

(Ik + a)

(a+ �L)nk@�k(� )

@� j�

mXk=1

@pk@� j

dk(p(� )) + n�jq�j (� )

!:

Denote the negative of the matrix M�1 by

B = � (M)�1 =

0BB@b11 b12 ::: b1mb21 b22 ::: b2m::: ::: ::: :::bm1 bm2 ::: bmm

1CCA :where Xk(� ) denotes total home production of good k, Xk(�) = nkqk(�). That is, we replicate the result(12) in Gawande, Krishna and Robbins (2004), up to the foreign lobbies component and the approximationthey use ( @pk

@�kdk(p(� )) � n�kq�k(� )).


Lemma 3 implies that each element of B is nonnegative, bij � 0. Multiplying equation (14)by B yields

� i =

mXj=1

bij

mXk=1

(Ik + a)

(a+ �L)nk@�k(� )

@� j�

mXk=1

@pk@� j

dk(p(� )) + n�jq�j (� )

!; (15)

which is equivalent to the system (10).Let us analyze the system (15). First, we note that the �rst-best trade tari¤s are given

by equation

�0i =mXj=1

bij

mXk=1

nk@�k(� )

@� j�

mXk=1

@pk@� j

dk(p(� )) + n�jq�j (� )

!:

As above, we �nd that if all industries are organized and every voter belongs to some lobby(�L = Ij = 1 8j), the equilibrium protection rates are �rst-best.

However, unlike the case with a separable utility (system (11)), it does not imply thatthe non-organized industries are always underprotected and the organized industries areoverprotected. Indeed, we see that in the presence of substitutability between products,the trade tari¤s are directly a¤ected by the sensitivity of supply and demand in the otherindustries and the organizational status of these industries. That is, the terms correspondingto the reaction of the other industries to the increase in � i, directly appear in the tari¤equation for industry i: In particular, the negative e¤ect of industry k being organizedon industry i0s protection is weaker if industries i and k produce substitutes. To see this,compare equations (11) and (15) for industry i. First, assume that the goods are independent(� = 0), so that the tari¤ for industry i is given by equation (11) for j = i. Other things equal,industry k getting organized reduces the protection for industry i only through an increasein �L (as the corresponding factor (Ii + a)=(a + �L) decreases and ni (@�i(� )=@� i) > 0 ).Now, instead, consider the case when the goods are substitutes and the tari¤ for industry i isgiven by equation (15)). As above, industry k getting organized reduces the industry i tari¤through higher �L in factors (Is + a)=(a+ �L) for all s 2 f1; :::;mg, as @�s(�)=@� j � 0 andbij � 0 for all i; j. However, it also has a positive e¤ect on industry i�s protection throughan increase in Ik from 0 to 1 in factor (Ik+a)=(a+�L). It is unclear which e¤ect dominates,but this argument suggests that the overall negative impact on protection of good i resultingfrom industry k being organized is weaker in case of substitutability between the goods.

The intuition behind this e¤ect is straightforward: in the absence of substitution, thatis, if the utility function is quasi-linear and separable in goods i = 1; :::;m, the consumptionof each non-numeraire good is fully determined by the price of this good only. As a result,each organized group has two goals. First, it tries to raise the trade tari¤ in its own sector,which increases its market share, the own good price and, thus, the pro�t of the lobby. Atthe same time, it attempts to reduce the tari¤s on all other goods it consumes, as lowertari¤s entail lower consumption prices. However, in the presence of substitution between thegoods, such a lobbying strategy may cause consumers to switch consumption from the mosthighly protected goods to less protected ones. To limit substitution, organized industriestend to apply more "moderate" lobbying strategies. That is, they try to maintain a balancebetween decreasing the price of the other goods and increasing its own price.

4. The level of protection 11

This e¤ect is best understood in case the ownership of industry k is very highly concen-trated, that is, the share �k of the population entitled to its pro�t is zero. Then, the totalshare of the population in the organized industries �L does not change when industry k be-comes organized. In this case, the e¤ect of other industries�price change on lobby k�s welfareis negligible, so that they have no consumer welfare gain from the trade interventions in othersectors. Therefore, in the absence of substitutability, if they get organized, it does not in�u-ence the protection of any other industry j 6= k (indeed, as �L does not change, equations(11) for tari¤s in industries j 6= k are una¤ected). However, with substitutability betweenthe goods, industry k still lobbies for additional protection of all substitute-producing indus-tries because it is concerned with maintaining demand for its own good. That is, it increasesindustry i protection, as the term

mXj=1

bijIk

(a+ �L)nk@�k(� )

@� j=

mXj=1

bij1

(a+ �L)nk@�k(� )

@� j

entering the equation for the equilibrium tari¤ � i is positive.In GH, the interests of di¤erent industry groups are opposed to each other. Here, we

see that non-organized industries may bene�t from the contributions of organized ones by"exploiting" the demand properties. We return to this discussion in the subsequent sectionsand show that this e¤ect caused by substitutability of products can lead to free-riding in thelobbying behavior.

The above results are obtained for Cournot competition. However, it is easily seenthat they also hold if we keep the assumption of linear demand and constant marginalcosts, but allow �rms to compete in prices in a Bertrand-fashion. Also in this case does ahigher tari¤ in industry i convert into higher prices and demand shifting away from sectori, while higher tari¤s in other sectors attract more demand into sector i. Indeed, prices arestrategic compliments and therefore, they increase in each sector�s tari¤s. From the pro�tmaximization problem it follows that, in equilibrium, higher domestic prices entail higherdomestic outputs

qi = (pi � ci)��@qi@pi

�;

and the same is true for foreign output in sector i for a given tari¤ � i

q�i = (pi � ci � � i)��@qi@pi

�: (16)

Therefore, domestic output increases in each sector�s tari¤ and foreign output increases inall other sectors� tari¤s. As the sensitivity of the foreign price with respect to the owntari¤ is never above one (an increase in industry i�s tari¤ is fully captured by an increase inprice only in case domestic and foreign goods in industry i are perfect substitutes), formula(16) implies that foreign output is decreasing in its own tari¤. It can be shown that underreasonable assumptions, the corresponding matrix M is invertible and thus the qualitativeresults do not change.

4 The level of protection

We are not ready to investigate whether, all else equal, more substitutability in consumerpreferences causes less protection. To make the analysis tractable, we assume that there is


only one domestic and one foreign �rm operating in each imperfectly competitive sector, sothat the respective non-numeraire good k is supplied by a duopoly, i.e.,

nk = n�k = 1;

Nk = 2:

Furthermore, we limit ourselves to the case of two non-numeraire goods, m = 2, and assumethe marginal costs of production to be equal across sectors, c1 = c2: We concentrate on theinterior solutions.7

In this case, the pro�t maximization problem of each domestic �rm takes the form

maxqk

�k = qk�1� (qk + q�k)� �

�q�k + q

��k�� cqk; k = 1; 2:

Similarly, each foreign �rm solves

maxqk

��k = q�k

�1� (qk + q�k)� �

�q�k + q

��k�� (c+ �k) qk; k = 1; 2:

>From the �rst-order conditions, it follows that for a given vector of trade tari¤s � =(�1; �2),in an interior Nash equilibrium each domestic �rm produces

qk =(1� c) (3� 2�) +

�3� 2�2

��k + ��k

9� 4�2 ; (17)

while each foreign �rm produces

q�k =(1� c) (3� 2�)� 2

�3� �2

��k + ��k

9� 4�2 : (18)

The pro�t in each case is determined by equality (4).

Substituting expressions (17), (18) and (4) into the system (10), we can solve for theequilibrium trade tari¤s. If none of the industries is organized, so that �L = I1 = I2 = 0,the government implements the �rst-best policy and imposes import taxes

�01 (�) = �02 (�) =

(1� c)� + 3

: (19)

Now, consider the case when only industry 1 is organized and it represents a proportion� of the total population. The resulting trade tari¤ for industry 1 is given by

�1 (�; �; a) = (1� c)�4 (a+ 1) (a+ 2�)�3 + 4

��2 � 3a�� 2a� 3a2 � 3�

��2

��9a2 + 26a�+ 10a+ 13�2 + 14�

��

+(3a+ �+ 2) (9a+ 11�)] ��4 (a+ 2�� 1) (a+ 2�)�4 (20)

+�14a� 118a�� 45a2 � 81�2 + 22�

��2

+(9a+ 11�� 2) (9a+ 11�)]�1 ;

7 It will be shown in the proof of Proposition 4 that a necessary condition for an interior solution isa+ 3� � 2.

4. The level of protection 13

and the tari¤ for industry 2 is determined by

�2 (�; �; a) = (1� c)�4a (a+ 2�� 1)�3 � 4

�a+ 2�+ 3a�� 2 + 3a2

��2

+ (18a� 26a�� 9a2 � 13�2 + 14�)�+(3a+ �) (9a+ 11�� 2)] �

�4 (a+ 2�� 1) (a+ 2�)�4 (21)

+�14a� 118a�� 45a2 � 81�2 + 22�

��2

+(9a+ 11�� 2) (9a+ 11�)]�1 ;

(see the Appendix for the derivation of equations (19), (20) and (21)).We would like to show that the increase in the degree of competition in the product market

(represented by the increase in the degree of substitution between the products) reduces theincentives for the lobby to raise its own price and/or lower the prices of the other goods.Hence, it results in a more moderate equilibrium trade protection for the organized interestgroup.

As shown by equation (19), an increase in the degree of substitution decreases the pro-tection level even in the absence of lobbying �the �rst-best trade tari¤s decline in �. Theintuition is as follows: A domestic import tax in sector i is aimed at increasing the marketshare of the domestic �rm.8 As the degree of substitutability increases, the e¤ect of the tradetax in sector i also extends to the �rms in sector �i, thereby increasing their output andimproving their market position. Hence, due to the competition arising from substitutability,the overall e¤ect of a tax in sector i on the �rm in sector i becomes weaker, leaving less roomfor strategic trade policy. When examining the in�uence of the degree of substitutability onlobbying, we want to abstract from this e¤ect. We do it by analyzing how the trade tari¤sin a lobbying equilibrium di¤er from the �rst-best trade policy. More precisely, we study theratios

Ti (�; �; a) = � i (�; �; a) =�0i (�) ; i = 1; 2;

which we henceforth refer to as relative protection, and their response to the change in thedegree of substitution �:

Proposition 4 If industry 1 is organized, while industry 2 is not, the equilibrium relativeprotection of the organized industry decreases with the degree of substitution:

dT1 (�; �; a)

d�< 0:

For the non-organized industry, the e¤ect is the opposite �the equilibrium relative protectionincreases with the degree of substitution:

dT2 (�; �; a)

d�> 0:

Proof. See the Appendix.It is worth noting that the result of Proposition 4 does not necessarily imply that the

in�uence-driven protection rates will converge to the socially-optimal level as the degreeof substitutability increases. In fact, for a larger �, the organized industry is interested

8Subject to the respective loss in consumer welfare and the change in the tax revenues.


in reducing the di¤erence between the price of its own good and the substitute, not inachieving the �rst-best outcome. For � = 1, the goods become indistinguishable from theconsumer�s point of view, i.e., industries 1 and 2 face a joint demand for these goods. Hence,the foreign �rms and sector 1 and sector 2 become identical competitors from the point ofview of the domestic organized industry 1, so it lobbies for the same trade tari¤ in bothindustries. However, depending on the concentration of industry 1 (that is, of the share ofpopulation �1 = � that has claims to its pro�t) this tari¤ can di¤er from the �rst-best levelin either direction. If � = 1=2, industry 1�s members are exactly representative of the entirepopulation � they own as much of the �rm�s pro�t claims 9 as does the average person.Therefore, the tari¤ for which the industry is lobbying perfectly matches the �rst-best tari¤.If instead industry 1 is more concentrated, so that � < 1=2, it cares about the domesticpro�ts, as opposed to consumer welfare, more than does the average person. Therefore, inthis case, the equilibrium protection rates exceed the socially optimal ones. Similarly, if� > 1=2, the resulting equilibrium protection falls below the socially optimal level.

Still, if the degree of substitution is not too high, the non-organized industry is alwaysunderprotected as compared to the �rst-best trade tari¤, because

T2 (0; �; a) = 3(3a+ �)

(9a+ 11�)= 1� 8 �

9a+ 11�< 1;

and the relative protection rate changes continuously. The organized industry achieves morethan the �rst-best protection, as

T1 (0; �; a) = 3(3a+ �+ 2)

(9a+ 11�� 2) = 1 +8 (1� �)

9a+ 11�� 2 > 1:10

Therefore, we have the following corollary:

Corollary 5 If the degree of substitutability is not too high, an increase in � shifts thein�uence-driven protection rates towards the socially optimal levels.

We have shown above that competition in the lobbying market can reduce the deviationof the in�uence-based protection from the �rst-best level. For example, when every personowns shares of one or the other industry and both industries actively participate in lobby-ing, the economy ends up with the socially-optimal trade tari¤s. Corollary 5 demonstratesthat competitive pressure in the product market (resulting from the rise in the degree ofsubstitution) can work in the same direction.

This result suggests a new explanation for a puzzle commonly observed in the empiricalstudies of GH theory. Most studies report extremely low estimates of the weight governmentputs on campaign contributions relative to social welfare.11 That is, the government behavesalmost as a social welfare maximizer and trade policy deviates surprisingly little from the�rst-best, which has caused a concern about the empirical signi�cance of the GH model. Forexample, Gawande and Krishna (2001) write that "..it is enough to cast doubt on the value

9Note that in case of perfect substitutability, domestic �rms in sectors 1 and 2 are exactly identical as aretheir pro�t functions.10As shown in the proof of Proposition 4, the necessary condition for the interior solution is a + 3� > 2,

which implies that 9a+ 11�� 2 > 0:11See e.g. Goldberg and Maggi (1999) and Gawande and Bandyopadhyay (2000).

5. Substitutability and lobbying activity 15

of viewing trade policy determination through this political economy lens". Recently, severalpapers have modi�ed the GH model in order to resolve the puzzle. This research primarilyemphasizes the idea of a lobbying competition. Gawande and Bandyopadhyay (2000) intro-duce political competition between the producers of intermediate and �nal goods, Gawandeand Krishna (2005) go even further by bringing in the cross-sectoral connections through theinput-output matrix, and Gawande, Krishna and Robbins (2004) introduce counter-lobbyingby foreign organized groups. This paper provides an alternative explanation, suggesting thatsmaller deviations from the �rst-best trade policy may result from the weaker incentives tolobby, due to product substitutability. Indeed, all studies cited above are based on 3-4 digitSIC industry data, which suggests at least some degree of substitutability between the prod-ucts of the di¤erent industries. Therefore, an organized industry�s lobbying decisions maywell be a¤ected by the potential demand shift e¤ects.

5 Substitutability and lobbying activity

So far, as in the original GH model, we assumed an exogenous lobby group structure. Butwhy are only some industries organized? In the previous section, we demonstrated thatif industry i is organized, the import tax for the substitute product �i increases with thedegree of substitutability. Hence, a su¢ ciently high degree of substitutability may entail asituation where a non-organized industry becomes protected without paying for it. That is,there may be free-riding in lobbying.

Let us illustrate this reasoning with an example. We continue to consider the 2-sector-2-�rm model of the previous section but impose several additional assumptions. First, assumethat the non-numeraire products are perfect substitutes, � = 1. Second, assume that theindustries have the same size �1 = �2 = �. We allow some of the voters to own no claims toany of the industries�pro�t but the labor only, so � � 1=2. To study the free-riding behavior,we extend the existing game by allowing for an initial stage 0 of costless lobby formation.In this stage, industries simultaneously decide whether they are getting organized. Theindustries deciding to form a lobby participate in the standard lobbying game in stage 1:

Using backward induction, we start by discussing stage 1. Compare the equilibriumoutcomes under the two regimes: (i) when only industry 1 is organized and (ii) when bothindustries 1 and 2 are organized. Once more, we concentrate on interior solutions.

If only industry 1 is organized, the trade tari¤s in this economy are determined byequations (20) and (21) for � = 1. This entails the equilibrium tari¤s

e�1 (1; �; a) = e�2 (1; �; a) � e� = (1� c)4

5a+ �+ 2

(5a+ 7�� 1) : (22)

As mentioned above, the import taxes on both goods are the same, even though only oneof the industries is organized. Under perfect substitutability, industry 1 treats the foreignproducers of both good 1 and good 2 as similar Cournot competitors. The symmetry of thesetting implies that the preferred trade policy of the organized lobby does not di¤er withrespect to the foreign �rms in sectors 1 and 2. Thus, the government also sets identical tradetari¤s for these two goods.

In this equilibrium, the non-organized industry 2 gains more from trade protection thanthe organized (and contributing) industry 1. Indeed, as the industries are identical and so


are their trade tari¤s, the gross welfare levels of industries 1 and 2, that is, the welfare beforethe lobbying contributions are paid, coincide. Formally,

W1(e� ) =W2(e� );where e� denotes the tari¤ vector in this equilibrium, e� = (e�1;e�2). However, the trade tari¤sin this equilibrium di¤er from the �rst best level, which implies that the organized industrymakes a positive contribution to the government. Therefore, the net welfare level of industry1 is below the net welfare level of industry 2:

V1(e� ) =W1(e� )� C1(e� ) < W2(e� ) = V2(e� ):In this regime, industry 2 bene�ts from the trade policy while not paying for it.

Now, consider the case when the domestic producers of both good 1 and good 2 areorganized. If they both participate in lobbying, solving the equation (10) for � = 1 and�L = 2� � 1 yields the equilibrium import tari¤s

ee�1 (1; �; a) = ee�2 (1; �; a) � ee� = (1� c)4

5a+ 2�+ 4

5a+ 14�� 2 ; (23)

(see the Appendix). Note that with two industries actively lobbying, the import tax is furtherfrom the �rst-best level. Formally,

ee� � e� � �0 (� = 1) = 1� c4:12 (24)

Indeed, due to the perfect substitutability (� = 1) and equal size (�1 = �2 = �), industries1 and 2 have exactly the same preferences. Hence, when they both actively participate inlobbying, the resulting policy is more biased towards the interest groups�preferred importtax.

We are interested in the trade-o¤ between the costs and bene�ts of this bias for bothindustry 1, which is lobbying in either equilibrium, and for industry 2, which only contributesin the second equilibrium. In other words, we want to compare the lobbies�payo¤s betweenthese two equilibria. We consider the case when organized industries play globally truthfulstrategies, that is when the contributions of lobby i are (globally) equal to the excess of thelobby�s welfare over a certain threshold Bj ,

Cj(e� ;Bj) = max [0;Wj(e� )�Bj ] :All truthful equilibria are also locally truthful, so in our argument we can rely on the resultsobtained in equations (22) and (23).

Note that the tari¤ given by equation (23) corresponds to the equilibrium when bothorganized industries actively participate in lobbying. In the original GH setting, the inter-ests of di¤erent industries are opposed to each other so, if organized, each industry indeedprefers to buy the protection. However, in the presence of substitutes, this outcome is notnecessarily unique. Indeed, if both industries are allowed to lobby, but each of them can getprotected without paying for it, the game may admit equilibria where only one of two orga-nized industries is active. Alternatively, there can be equilibria where both industries lobbybut make di¤erent contributions. In what follows, we concentrate on symmetric truthfulequilibria, which seems to be natural given the symmetry of the setting.


Denote the truthful equilibrium when only industry 1 is organized by E1 and a symmetricequilibrium when both industries are organized by E1&2. We start by evaluating the amountof contributions and the government surplus in these two equilibria.

Lemma 6 a) The government is equally well o¤ in E1 and E1&2: b) The total contributionsto the government are greater in E1&2 than in E1.

Proof. To understand these results, we need to calculate truthful equilibrium contributionsand net welfare levels. In a truthful equilibrium, each lobby j chooses a scalar anchor Bj sothat the government would be just indi¤erent between choosing the equilibrium policy � andthe policy ��j , which is de�ned as the policy chosen by the government if the contributionsof lobby j were zero:

��j = argmax�2T

Xi2L;i6=j

Ci(� ;Bi)+aW (� ): (25)

Then, the contribution of lobby j solvesXi2L;i6=j

Ci(��j ;Bi)+aW (�

�j) =Xi2L

Ci(� ;Bi)+aW (� ): (26)

As a consistency check, we should have each lobby j making no contribution at the tari¤vector ��j

Wj(��j) � Bj ; (27)

as otherwise it can increase its reservation utility Bj at no cost.We are now ready to apply this procedure. First, we establish that in E1, the government

receives exactly the payo¤ it would get in the �rst-best equilibrium. Indeed, if industry 1were to contribute zero, the government would get no contributions at all, and would thusmaximize the gross social welfare

e��j = argmax�2T

aW (� ) = � 0: (28)

This observation and the government indi¤erence condition (26) immediately imply that insuch an equilibrium, the government gets exactly G(� 0):

Now, we show that the government receives exactly as much in E1&2. Indeed, by theconstruction of the equilibrium, condition (27) holds and industry 1 does not make any

contribution at the tari¤ vector ee��1:C1(ee��1;B1) = max h0;W1(ee��1)�B1i = 0: (29)

As the goods are perfect substitutes and the two industries are exactly alike, their gross

welfare is the same under policy ee��1: Moreover, as the equilibrium E1&2 is symmetric, theanchors of two industries are the same (B1 = B2). Therefore, industry 2 does not contribute

anything at the tari¤ vector ee��1 either. Hence, if the contributions of lobby 1 were zero, thegovernment would choose the free-trade policy

ee��1 = argmax�2T

aW (� ) = � 0: (30)


The same result holds for trade policy ee��2: Condition (26) immediately implies that in suchan equilibrium, the government�s payo¤ is the same as in the �rst-best equilibrium withoutany lobbying, which proves part a) of the Lemma.

Part b) follows from part a) and our observation (24). Indeed, when both industriesare organized, the equilibrium tari¤s are higher than in the case when only one industryis lobbying, and thus further from the �rst-best outcome. Thus, the government requireshigher contributions to compensate for the larger social welfare loss.

Thus, we see that the equilibrium with two organized industries provides the lobbieswith higher gross welfare, but requires additional lobbying contributions. However, thesecontributions are now paid by two lobbies, as compared to the equilibrium with a singleorganized industry. So lobby 1, which was active in both equilibria, bene�ts more (or losesless) than lobby 2; as it can now share the costs. Still, it is not clear whether either of themactually wins or loses in the equilibrium E1&2. This question is answered in the followingproposition.

Proposition 7 a) Lobby 1 has a higher net welfare in E1&2 than in E1 for any admissibleparameter values. b) Lobby 2 has a lower net welfare in E1&2 than in E1, if and only if� > 1=7.

Proof. We start by calculating lobbying contributions to the government in equilibrium E1.From (26) and (28), it follows that

C1

�e� ; eB1�=aW �� 0�� aW (e� ):

Hence, the net welfare of lobby 1 is

V1(e� ) � fB1 =W1(e� )� C1(e� ;fB1) =W1(e� )� aW (� 0) + aW (e� ): (31)

As industry 2 is not organized, its net welfare is equal to its gross welfare,

V2(e� ) =W2(e� ): (32)

In equilibrium E1&2, conditions (26), (29) and (30) determine the aggregate contributions

C1(ee� ;ffB1) + C2(ee� ;ffB2) = aW (� 0)� aW (ee� ):As this equilibrium is symmetric, the anchors of both lobbies are the same and so is theirgross welfare. Hence, their contributions are also equal and comprise half of the aggregateamount

C1(ee� ;ffB1) = C2(ee� ;ffB2) = 1

2

�aW (� 0)� aW (ee� )� :

The payo¤ of either lobby group is thus

Vi(ee� ) � ffBi =Wi(ee� )� Ci(ee� ;ffBi) =Wi(ee� )� 12

�aW (� 0)� aW (ee� )� ; i = 1; 2: (33)

So in order to determine whether industry 1 gains from the equilibrium with two lobbies,payo¤s (31) and (33) must be compared. It can be shown (see the Appendix) that

V1(ee� )� V1(e� ) = 9a

20

(1� 2�)2 (1� c)2

(5a+ 7�� 1) (5a+ 14�� 2) � 0: (34)


Similarly, the welfare di¤erence of industry 2 is given by payo¤s (32) and (33). Simplifyingthe expression (see the Appendix), we get

V2(ee� )� V2(e� ) = 9a

20

(1� 2�)2 (1� c)2

(5a+ 14�� 2) (5a+ 7�� 1)2(1� 7�)

�< 0; � < 1=7;

� 0; � � 1=7: : (35)

So as long as the size of the industry is su¢ ciently large, it loses from participating in lob-bying. The intuition behind this result is as follows: if the industries are very concentrated(� = 0), they highly bene�t from an increase in protection as the loss in their members�consumer welfare resulting from higher prices is negligible. With decreasing ownership con-centration (higher �), more and more consumers in the industry lose from the price increasewhich results in a decrease in protection rates and industry welfare. And it may be the casethat the gain of the industry from increased protection (due to this industry participationin lobbying) is not high enough to cover the necessary contribution for this increase.

We characterized the outcome of the symmetric truthful equilibrium when both industriesare organized, given its existence. The following lemma establishes that it indeed exists.

Lemma 8 In the lobbying game with perfect substitutability (� = 1) and industries of iden-tical size, there exists a symmetric truthful equilibrium.

Proof. See the Appendix.Now, we turn to the lobby formation stage. In this stage, each industry decides whether it

gets organized (O) and buys in�uence in the next stage of the game, or stays non-organized(N) and remains passive in the lobbying game. The original GH setup entails a singleequilibrium of a type (O,O): getting organized is a dominant strategy in that game as di¤erentlobbies�interests are strictly opposed to each other. The same happens in our setting if theindustry�s ownership structure is very concentrated (low �). Then, again, getting organizedis a dominant strategy and a single (O,O) equilibrium emerges.

However, if an industry has a dispersed ownership structure (higher �), it prefers tocommit not to lobby as long as the substitute industry will be lobbying. That is, the lobbyformation stage is a �chicken game�, where each industry prefers to be organized when theother is not and vise versa. The payo¤s of the game are

Ind.1 / Ind.2 organized non-organizedorganized (A;A; ) (B;C)

non-organized (C;B) (D;D)

where

A = Vi(ee� ) =Wi(ee� )� 12

�aW (� 0)� aW (ee� )� ;

B = V1(e� ) =W1(e� )� aW (� 0) + aW (e� );C = V2(e� ) =Wi(e� );D = Wi(�

0):

As we have shown, C > A > B > D: This game has two pure strategy Nash equilibria,(O,N) and (N,O), and one mixed strategy equilibrium. The outcome (O,O) is no longer an


equilibrium of the game. In other words, in the presence of substitutability, fewer industriesget organized, and lobbying becomes less intense. This discussion is summarized in thefollowing corollary.

Corollary 9 In the game extended by the participation decision stage, a dispersed ownershipstructure weakens the incentives of the industries to get organized which, in turn, leads to aless intensive lobbying process.

Note that aggregate net welfare of industries 1 and 2 is larger in E1&2 than in E1, as

V1(ee� ) + V2(ee� )� (V1(e� ) + V2(e� )) = 9

4

a2 (1� 2�)2 (1� c)2

(5a+ 14�� 2) (5a+ 7�� 1)2> 0:

The free-riding problem in lobbying can thus be summarized as follows: product substi-tutability produces a positive inter-industry externality from protection which, in turn, leadsto dilution of the incentives to organize smaller protection than the industries desire. Bycontinuity, this result also extends to markets with high, but not perfect substitutability.

6 Conclusion

This paper studies the impact of demand linkages on the determination of trade policy, andthe intensity of inter-industry lobbying competition. We �nd that product substitutabilityboth weakens industries�incentives to get organized and their lobbying incentives when theyare organized. The �nding may explain why empirical investigations based on the "Protectionfor Sale" model have suggested that the government is almost exclusively concerned withwelfare rather than contributions by lobbies.

REFERENCES 21

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[9] Eaton, Jonathan and Gene Grossman, 1986, "Optimal trade and industrial policy underoligopoly", Quarterly Journal of Economics 101, 383�406.

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[11] Findlay, Ronald and Stanislaw Wellisz, 1982, �Endogenous Tari¤s, The Political Econ-omy of Trade Restrictions and Welfare�, in J.N. Bhagwati and T.N. Srinivasan, eds.,Import Competition and Response, Chicago, University of Chicago Press, 223-234.

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A. Appendix 23

A Appendix

A.1 Derivation of equation (10)

We begin by simplifying equation (9):

(a+ �L)r"mXk=1

�kn�kq�k(� )�

mXk=1

pkdk(p(� )) + U(d1(p(� )); ::; dm(p(� )))

#

+mXk=1

(Ik + a)r�k(� ) = 0:

Start by evaluating

@

@� j

"mXk=1

�kn�kq�k(� )�

mXk=1

pkdk(p(� )) + U(d1(p(� )); :::; dm(p(� )))

#: (36)

As

@

@� j

mXk=1

�kn�kq�k(� )

!= n�jq

�j (� ) +

mXk=1

�kn�k

@q�k(� )

@� j;

@

@� j

�

mXk=1

pkdk(p(� ))

!=

�

mXk=1

@pk@� j

dk(p(� ))�mXk=1

pk@dk(p(� ))

@� j

!;

and

@

@� j

�U(d1(p(� )); ::; dm(p(� )))

�=

mXi=1

@U(d1(p(� )); ::; dm(p(� )))

@ (di(p(� ))� @di(p(� ))

@� j

=mXi=1

pi@di(p(� ))

@� j;

expression (36) is equivalent to

@

@� j

"mXk=1

�kn�kq�k(� )�

mXk=1

pkdk(p(� )) + U(d1(p(� )); ::; dm(p(� )))

#=

n�jq�j (� ) +

mXk=1

�kn�k

@q�k(� )

@� j�

mXk=1

@pk@� j

dk(p(� ))�mXk=1

pk@dk(p(� ))

@� j+

mXi=1

pi@di(p(� ))

@� j= n�jq

�j (� ) +

mXk=1

�kn�k

@q�k(� )

@� j�

mXk=1

@pk@� j

dk(p(� )):

Thus, each line of condition (9) takes the form

(a+ �L)

n�jq

�j (� ) +

mXk=1

�kn�k

@q�k(� )

@� j�

mXk=1

@pk@� j

dk(p(� ))

!+

mXk=1

(Ik + a)nk@�k(� )

@� j= 0;

which can be rewritten as

�mXk=1

�kn�k

@q�k(� )

@� j=

mXk=1

(Ik + a)

(a+ �L)nk@�k(� )

@� j�

mXk=1

@pk@� j

dk(p(� )) + n�jq�j (� ): (37)


Substituting de�nition mk = n�kq�k into expression (37) yields

�mXk=1

�k@mk(� )

@� j=

mXk=1

(Ik + a)

(a+ �L)nk@�k(� )

@� j�

mXk=1

@pk@� j

dk(p(� )) + n�jq�j (� ):

A.2 Proof of Lemma 1.

The �rst-order condition associated with the pro�t maximization problem of any of the nksymmetric domestic �rms in sector k (2) is given by

1� nkqk � n�kq�k � �Xs 6=k

(nsqs + n�sq�s) = ck + qk: (38)

Similarly, the �rst-order condition for any of the n�k symmetric foreign �rms is given by

1� nkqk � n�kq�k � �Xs 6=k

(nsqs + n�sq�s) = ck + q

�k + �k: (39)

It follows thatqk = q

�k + �k 8k: (40)

Let aggregate output in industry k be denoted

Qk = nkqk + n�kq�k = Nkqk � n�k�k: (41)

Then (38) for each k takes the form:

1��1 +

1

Nk

�Qk � �

Xl 6=kQL = ck �

n�kNk�k; (42)

and the entire system of equations for all k transforms into0BB@1 + 1=N1 � ::: �

� 1 + 1=N2 ::: �::: ::: ::: :::� � ::: 1 + 1=Nm

1CCA0BB@Q1Q2:::Qm

1CCA =

0BBB@1� c1 � n�1

N1�1

1� c2 � n�2N2�2

:::

1� cm � n�mNm�m

1CCCA : (43)

Let us study the matrix

S =

0BB@1 + 1=N1 � ::: �

� 1 + 1=N2 ::: �::: ::: ::: :::� � ::: 1 + 1=Nm

1CCA :First, the determinant of S is positive:

detS =N1 + 1

N1

mYi=2

�Ni + 1

Ni� �

�+ �

mXi=2

24 mYj=1;j 6=i

�Nj + 1

Nj� �

�35 > 0:Therefore, matrix S is invertible. Denote its inverse by R = S�1. Then, the elements ofmatrix R can be written

R11 =1

detS

0@N2 + 1N2

mYi=3

�Ni + 1

Ni� �

�+ �

mXi=3

24 mYj=2;j 6=i

�Nj + 1

Nj� �

�351A > 0;

A. Appendix 25

Rkk =1

detS

0@N1 + 1N1

mYi=2;i6=k

�Ni + 1

Ni� �

�(44)

+�mX

i=2;i6=k

24 mYj=1;j 6=i;j 6=k

�Nj + 1

Nj� �

�351A > 0;

for 1 < k � m, and

Rki =1

detS

0@�� mYj=1;j 6=i;j 6=k

�Nj + 1

Nj� �

�1A � 0:

That is, matrix R is symmetric, its diagonal elements are positive, and the o¤-diagonalelements are non-positive. Therefore, system (43) can be rewritten as

0BB@Q1Q2:::Qm

1CCA = R

0BBB@1� c1 � n�1

N1�1

1� c2 � n�1N2�1

:::

1� cm � n�mNm�m

1CCCA ; (45)

and we see that

@Qi@� j

�< 0; i = j

> 0; i 6= j : (46)

Moreover, from (41), it follows that

0BB@q1q2:::qm

1CCA =

0BB@(Q1 + n

�1�1) =N1

(Q2 + n�2�2) =N2:::

(Qm + n�m�m) =Nm

1CCA : (47)

So

@qk@�k

=1

Nk

�@Qk@�k

+ n�k

�=

1

Nk

�� n

�k

NkRkk + n

�k

�: (48)

Next we show that for any k

Rkk < Nk: (49)

>From formula (44), inequality (49) is equivalent to

0@N1 + 1N1

mYi=2;i6=k

�Ni + 1

Ni� �

�+ �

mXi=2;i6=k

24 mYj=1;j 6=i;j 6=k

�Nj + 1

Nj� �

�351A�Nk detS < 0:


Denote this di¤erence by D. It is equal to

D =

0BB@N1 + 1N1

mYi=2;i6=k

�Ni + 1

Ni� �

�+ �

mXi=2;i6=k

2664 mYj=1;

j 6=i;j 6=k

�Nj + 1

Nj� �

�37751CCA�Nk detS

=N1 + 1

N1

mYi=2;i6=k

�Ni + 1

Ni� �

�+ �

mXi=2;i6=k

2664 mYj=1;

j 6=i;j 6=k

�Nj + 1

Nj� �

�3775�Nk

0@N1 + 1N1

mYi=2

�Ni + 1

Ni� �

�+ �

mXi=2

24 mYj=1;j 6=i

�Nj + 1

Nj� �

�351A :By rearranging terms and simplifying, we rewrite it as

D = Nk (� � 1)

0BB@N1 + 1N1

mYi=2;i6=k

�Ni + 1

Ni� �

�+ �

mXi=2;i6=k

2664 mYj=1;

j 6=i;j 6=k

�Nj + 1

Nj� �

�37751CCA

��mY

j=1;j 6=k

�Nj + 1

Nj� �

�:

As � 2 [0; 1], D < 0; proving that Rkk < Nk as desired.

Applying this result to equality (48), we conclude that

@qk@�k

=1

Nk

�� n

�k

NkRkk + n

�k

�>n�kNk

��NkNk

+ 1

�= 0:

Equalities (46) and (47) imply that

@qk@� j

=1

Nk

@Qk@� j

> 0; k 6= j:

Finally, relations (40) and (41) suggest that0BB@q�1q�2:::q�m

1CCA =

0BB@(Q1 � n1�1) =N1(Q2 � n2�2) =N2

:::(Qm � nm�m) =Nm

1CCA :As a result, by using inequalities (46), we conclude that

@q�k@�k

=1

Nk

�@Qk

@�k� nk

�< 0

and@q�k@� j

=1

Nk

@Qk@� j

> 0; k 6= j:

A. Appendix 27

A.3 Proof of Lemma 3.

Equality (40) allows us to rewrite the �rst-order conditions(39) as

1� (Nk + 1) q�k � �Xs 6=kNsq

�s = ck + (nk + 1) �k + �

Xs 6=kns� s:

This can be rewritten in a matrix form as

0BB@1 +N1 �N2 ::: �Nm�N1 1 +N2 ::: �Nm::: ::: ::: :::�N1 �N2 ::: 1 +Nm

1CCA0BB@q�1q�2:::q�m

1CCA (50)

=

0BB@1� c11� c2:::

1� cm

1CCA�0BB@n1 + 1 n2� ::: nm�n1� n2 + 1 ::: nm�::: ::: ::: :::n1� n2� ::: nm + 1

1CCA0BB@�1�2:::�m

1CCA :Note that0BB@1 +N1 �N2 ::: �Nm�N1 1 +N2 ::: �Nm::: ::: ::: :::�N1 �N2 ::: 1 +Nm

1CCA =

0BBB@1+N1N1

� ::: �

� 1+N2N2

::: �

::: ::: ::: :::

� � ::: 1+NmNm

1CCCA0BB@N1 0 ::: 00 N2 ::: 0::: ::: ::: :::0 0 ::: Nm

1CCA :The �rst matrix in this product is matrix S de�ned above, and we have shown that it isinvertible. Clearly, the diagonal matrix

N =

0BB@N1 0 ::: 00 N2 ::: 0::: ::: ::: :::0 0 ::: Nm

1CCAis invertible as Ni > 0 for all i = 1; :::m. Denote its inverse by N�1. Therefore, equation(50) can be rewritten as0BB@

q�1q�2:::q�m

1CCA =N�1R

26640BB@1� c11� c2:::

1� cm

1CCA�0BB@n1 + 1 n2� ::: nm�n1� n2 + 1 ::: nm�::: ::: ::: :::n1� n2� ::: nm + 1

1CCA0BB@�1�2:::�m

1CCA3775 :

It follows that the matrix of the derivatives of the output of a single foreign �rm in sector kwith respect to the trade tari¤s in sector i is given by

�@q�k@� j

�= �N�1R

0BB@n1 + 1 n2� ::: nm�n1� n2 + 1 ::: nm�::: ::: ::: :::n1� n2� ::: nm + 1

1CCA :


As a result, the matrix of the derivatives of the aggregate import in sector k with respect tothe trade tari¤s is

M =

0BB@n�1 0 ::: 00 n�2 ::: 0::: ::: ::: :::0 0 ::: n�m

1CCA�@q�k@� j�

= �

0BB@n�1 0 ::: 00 n�2 ::: 0::: ::: ::: :::0 0 ::: n�m

1CCAN�1R

0BB@n1 + 1 n2� ::: nm�n1� n2 + 1 ::: nm�::: ::: ::: :::n1� n2� ::: nm + 1

1CCA :Note that0BB@n1 + 1 n2� ::: nm�n1� n2 + 1 ::: nm�::: ::: ::: :::n1� n2� ::: nm + 1

1CCA =

0BB@n1+1n1

� ::: �

� n2+1n2

::: �

::: ::: ::: :::� � ::: nm+1

nm

1CCA0BB@n1 0 ::: 00 n2 ::: 0::: ::: ::: :::0 0 ::: nm

1CCA :Clearly, as ni > 0, the inverse of 0BB@

n1 0 ::: 00 n2 ::: 0::: ::: ::: :::0 0 ::: nm

1CCAis a diagonal matrix with positive elements. Similarly to our derivation in the proof of Lemma1, the matrix

s =

0BB@n1+1n1

� ::: �

� n2+1n2

::: �

::: ::: ::: :::� � ::: nm+1

nm

1CCAis also invertible. Its determinant is positive and equal to

det s =n1 + 1

n1

mYi=2

�ni + 1

ni� �

�+ �

mXi=2

24 mYj=1;j 6=i

�nj + 1

nj� �

�35 > 0:Denote its inverse by r = s�1. Then, the elements of matrix r are given by

r11 =1

det s

0@n2 + 1n2

mYi=3

�ni + 1

ni� �

�+ �

mXi=3

24 mYj=2;j 6=i

�nj + 1

nj� �

�351A > 0;

rkk =1

det s

0@n1 + 1n1

mYi=2;i6=k

�ni + 1

ni� �

�+ �

mXi=2;i6=k

24 mYj=1;j 6=i;j 6=k

�nj + 1

nj� �

�351A > 0;

and

rki =1

det s

0@�� mYj=1;j 6=i;j 6=k

�nj + 1

nj� �

�1A � 0:

A. Appendix 29

Therefore, matrix M is invertible as a product of invertible matrices. It remains to showthat all elements of its inverse are non-positive, i.e., that

M�1 =

0BB@n1 0 ::: 00 n2 ::: 0::: ::: ::: :::0 0 ::: nm

1CCA�1

rSN

0BB@n�1 0 ::: 00 n�2 ::: 0::: ::: ::: :::0 0 ::: n�m

1CCA�1

= 0 : (51)

Lemma 10 The matrix rSN = 0.

Proof. As matrix SN is

SN =

0BB@1 +N1 �N2 ::: �Nm�N1 1 +N2 ::: �Nm::: ::: ::: :::�N1 �N2 ::: 1 +Nm

1CCA ;the diagonal element (k; k) of the matrix rSN is given by0@(1 +Nk)rkk + �Nk mX

i=1;i6=krki

1A :As the determinant of the matrix s = r�1 is positive, the sign of this element does notchange from multiplication by det s. Therefore, the sign of the diagonal element (k; k) of thematrix rSN is equal to the sign of the following expression:

det s�

0@(1 +Nk)rkk + �Nk mXi=1;i6=k

rki

1A= (1 +Nk)

0@n1 + 1n1

mYi=2;i6=k

�ni + 1

ni� �

�+ �

mXi=2;i6=k

24 mYj=1;j 6=i;j 6=k

�nj + 1

nj� �

�351A��2Nk

mXi=1;i6=k

mYj=1;j 6=i;j 6=k

�nj + 1

nj� �

�

> (1 +Nk)

0@n1 + 1n1

mYi=2;i6=k

�ni + 1

ni� �

�1A� �2Nk mYj=2;;j 6=k

�nj + 1

nj� �

�(52)

=

�(1 +Nk)

n1 + 1

n1� �2Nk

�0@n1 + 1n1

mYi=2;i6=k

�ni + 1

ni� �

�1A > 0;

where the inequality in (52) follows from � > �2, 1 +Nk > Nk and

mYj=1;j 6=i;j 6=k

�nj + 1

nj� �

�> 0:


Similarly, the sign of the o¤-diagonal element (k; j) is equal to the sign of the expression

det s�

0@(1 +Nj)rkj + �Nj mXi=1;i6=j;i6=k

rki + �Njrkk

1A= � (1 +Nj)�

mYi=1;i6=j;i 6=k

�ni + 1

ni� �

�� 2Nj

mXi=1;i6=k;i6=j

mYl=1;l 6=i;l 6=k

�nl + 1

nl� �

�

+�Nj

0@n1 + 1n1

mYi=2;i6=k

�ni + 1

ni� �

�+ �

mXi=2;i6=k

24 mYl=1;l 6=i;l 6=k

�nl + 1

nl� �

�351A=

�� (1 +Nj)

�n1 + 1

n1� �

�� 2Nj

�nj + 1

nj� �

�

+�Nj

�n1 + 1

n1� �

�+Nj

�n1 + 1

n1

��nj + 1

nj� �

��0@ mYi=2;i6=k;i6=j

�ni + 1

ni� �

�1A= �

�(Nj � nj)

n1(1� �) + 1n1nj

�0@ mYi=2;i6=k;i6=j

�ni + 1

ni� �

�1A � 0

Therefore, all elements of matrix rSN are nonnegative.Clearly, both 0BB@

n1 0 ::: 00 n2 ::: 0::: ::: ::: :::0 0 ::: nm

1CCA�1

and 0BB@n�1 0 ::: 00 n�2 ::: 0::: ::: ::: :::0 0 ::: n�m

1CCA�1

are positive diagonal matrices, so the product of these and a nonnegative matrix is oncemore non-negative. From Lemma 10, it immediately follows that matrix M�1 determinedby equality (51) is non-positive.

A.4 Derivation of equation (19).

The output levels of the foreign and domestic �rms in sectors 1 and 2 are given by equations(17) and (18). Using them, we obtain the following formulas for the tari¤ sensitivity of theimports

@mk(� )

@�k=�2�3� �2

�9� 4�2 ;

@mk(� )

@��k=

�

9� 4�2 ;

domestic pro�ts

A. Appendix 31

@�k(� )

@�k= 2

�3� 2�2

� (1� c) (3� 2�) + �3� 2�2� �k + ��k(9� 4�2)2

;

@�k(� )

@��k= 2�

(1� c) (3� 2�) +�3� 2�2

��k + ��k

(9� 4�2)2;

price

@pk(� )

@�k=

�3� 2�2

�9� 4�2 ;

@pk(� )

@�k=

�

9� 4�2 ;

and aggregate domestic consumption in sectors 1 and 2

dk(p(� )) =2(3� 2�) (1� c)� 3�k + 2��k

9� 4�2 :

Substituting these relations into the system (10) yields a linear system

2�3� �2

��k � ��k

=(Ik + a)

(a+ �L)2�3� 2�2

� (1� c) (3� 2�) + �3� 2�2� �k + ��k(9� 4�2)

+(I�k + a)

(a+ �L)2�(1� c) (3� 2�) +

�3� 2�2

��k + ��k

(9� 4�2)

��3� 2�2

� 2(3� 2�) (1� c)� 3�k + 2��k9� 4�2

��2(3� 2�) (1� c)� 3��k + 2��k9� 4�2 + (1� c) (3� 2�)� 2

�3� �2

��k + ��k;

k = 1; 2. Collecting terms and multiplying by a common factor, we get an equivalent system��2�3� 2�2

�2 (Ik + a)(a+ �L)

� 2��3� 2�2

� (I�k + a)(a+ �L)

� 76�2 + 16�4 + 99��k

��2�3� 2�2

� (a+ Ik)(a+ �L)

+ 2�(a+ I�k)

(a+ �L)+ 15� 4�2

��k (53)

= (1� c) (3� 2�)�3� 2� + 2� (a+ I�k)

(a+ �L)+ 2

�3� 2�2

� (a+ Ik)(a+ �L)

�;

which determines trade tari¤s �1 and �2.In the absence of organized lobbies, I1 = I2 = �L = 0. Therefore, (Ik + a)=(a+ �L) = 1

for k = 1; 2, and the system can be simpli�ed into two symmetric equations

��20� � 4�2 + 4�3 + 27

��k � � (7� 4�) ��k = (1� c) (3� 2�)2

for k = 1; 2. Solving this yields

�1 = �2 =(1� c)3 + �

:


A.5 Derivation of equations (20) and (21).

Formulas (20) and (21) follow from solving system (53) for I1 = 1; �L = �, and I2 = 0.

A.6 Proof of Proposition 4.

To show that the ratio �1(�;�;a)�01(�)

decreases with � for any (a; �) ; we proceed in six steps.1. Describe the necessary condition for the interior solution for the tari¤.2. Compose the ratio T (�; �; a) = �1(�;�;a)

�01(�); represent it as a ratio of two polynomials

T (�; �; a) = N(�;�;a)D(�;�;a) and show that both the numerator N (�; �; a) and the denominator

D (�; �; a) decline in �:3. Show that T (�; �; a) declines in � on [0; 1] i¤ the ratio of the derivatives of N (�; �; a)

and D (�; �; a) ; R (�; �; a) � @N(�;�;a)@� �@D(�;�;a)

@� is higher than T (�; �; a) for all � 2 [0; 1]and (a; �) delivering an interior solution.

4. Introduce an auxiliary linear function A (�; �; a) and show that R (�; �; a) � A (�; �; a)for any admissible (�; �; a) :

5. Show that A (�; �; a) � T (�; �; a) for any admissible (�; �; a) :6. From steps 3, 4 and 5, conclude that R (�; �; a) � T (�; �; a) :

A.6.1 Necessary condition for the interior solution for the tari¤

The industry 1 tari¤ �1 (�; �; a) is given by equation (20) and the �rst-best tari¤ (in theabsence of any lobbying) �01(�) is determined by equation (19). So the ratio of the lobbyingtari¤ to the �rst-best is

T (�; �; a) =�1 (�; �; a)

�01(�)=

(3 + �) [4 (a+ 1) (a+ 2�)�3 +�4�2 � 12a�� 8a� 12a2 � 12�

��2

��10a+ 14�+ 26a�+ 13�2 + 9a2

�� + (3a+ �+ 2) (9a+ 11�)]=

[4�4 (a+ 2�� 1) (a+ 2�) + �2�14a+ 22�� 118a�� 81�2 � 45a2

�+(9a+ 11�� 2) (9a+ 11�)]:

We are interested in an interior solution. A necessary condition is that the foreign �rmproduces a non-negative amount of the good:

q�(0; �1; �2) =(1� c)� 2�1

3� 0:

This condition is equivalent to

�1 (0; �; a) �(1� c)2

,

3a+ �+ 2

9a+ 11�� 2 �1

2,

a+ 3� � 2: (54)

We assume that the necessary condition for the interior solution (54) always holds.

A. Appendix 33

A.6.2 Ratio T (�; �; a) = N (�; �; a) = D (�; �; a).

Denote the numerator of T (�; �; a) by

N (�; �; a) = (3 + �) [4 (a+ 1) (a+ 2�)�3 +�4�2 � 12a�� 8a� 12a2 � 12�

��2

��10a+ 14�+ 26a�+ 13�2 + 9a2

�� + (3a+ �+ 2) (9a+ 11�)];

and the denominator by

D (�; �; a) = [4�4 (a+ 2�� 1) (a+ 2�) + �2�14a+ 22�� 118a�� 81�2 � 45a2

�+(9a+ 11�� 2) (9a+ 11�)];

so that

T (�; �; a) =N (�; �; a)

D (�; �; a):

Note that both N (�; �; a) and D (�; �; a) are decreasing in � for any a; �: Indeed

N (�; �; a) = 3 (3a+ �+ 2) (9a+ 11�)� 4��3a+ 5�+ 9a�+ 7�2

��2

�34a+ 50�+ 62a�+ �2 + 45a2

�+ 4�3

�a+ 3�+ 3a�+ �2

�+4�4 (a+ 1) (a+ 2�) :

The �rst derivative of N (�; �; a) with respect to � is

@N (�; �; a)

@�= �4

�3a+ 5�+ 9a�+ 7�2

�� 2�

�34a+ 50�+ 62a�+ �2 + 45a2

�+12�2

�a+ 3�+ 3a�+ �2

�+ 16�3 (a+ 1) (a+ 2�) :

The second derivative of N (�; �; a) is

@2N (�; �; a)

@�2= �2

�34a+ 50�+ 62a�+ �2 + 45a2

�+ 24�

�a+ 3�+ 3a�+ �2

�+48�2 (a+ 1) (a+ 2�) :

As (a+ 1) (a+ 2�) > 0; it is a convex quadratic parabola with the global minimum at

� = �a+ 3�+ 3a�+ �2

(a+ 1) (a+ 2�)< 0;

so that the second derivative of N (�; �; a) increases on � 2 [0; 1]: This derivative is negativeat � = 0

@2N (0; �; a)@�2

= �2�34a+ 50�+ 62a�+ �2 + 45a2

�< 0;

and can be of either sign at � = 1

@2N (1; �; a)@�2

= 2�2a+ 34�+ 22a�+ 11�2 � 21a2

�:

If it is negative at � = 1;@2N (1; �; a)

@�2< 0;


it is also negative at the entire � 2 [0:1], meaning that the �rst derivative of N (�; �; a)decreases in � 2 [0; 1]. If

@2N (1; �; a)@�2

> 0;

it changes sign only once, so the �rst derivative of N (1; �; a) with respect to �; @N(1;�;a)@� ;

�rst declines and then increases on � 2 [0; 1]: As

@N (0; �; a)

@�= �4

�3a+ 5�+ 9a�+ 7�2

�< 0

and@N (1; �; a)

@�= �2 (37a+ 9�+ 26) (a+ �) < 0;

we can conclude that @N(�;�;a)@� < 0 for any � 2 [0; 1], and, thus, that N (�; �; a) is decreasingin �.

Similarly,

D (�; �; a) = (9a+ 11�� 2) (9a+ 11�) +�2�14a+ 22�� 118a�� 81�2 � 45a2

�+ 4�4 (a+ 2�� 1) (a+ 2�) :

The �rst derivative of D (�; �; a) is

@D (�; �; a)

@�= 2�

�14a+ 22�� 118a�� 81�2 � 45a2

�+ 16�3 (a+ 2�� 1) (a+ 2�) :

The second derivative of D (�; �; a) is

@2D (�; �; a)

@�2= 2

�14a+ 22�� 118a�� 81�2 � 45a2

�+ 48�2 (a+ 2�� 1) (a+ 2�) :

It is once more a convex quadratic parabola with the global minimum at � = 0, so it increaseson � 2 [0; 1]: As it is negative at � = 1

@2D (1; �; a)

@�2= �2

�10a+ 26�+ 22a�� 15�2 + 21a2

�< 0;

it is thus negative at the entire segment � 2 [0; 1]. As a result, the �rst derivative @D(�;�;a)@�

decreases over �. As@D (0; �; a)

@�= 0;

we conclude that for any � 2 [0; 1];

@D (�; �; a)

@�� 0;

and thus, D (�; �; a) is decreasing in �:

A. Appendix 35

A.6.3 Necessary and su¢ cient condition for T (�; �; a) to decline in �

The derivative of T (�; �; a) is

@T (�; �; a)

@�=

@N(�;�;a)@� D (�; �; a)� @D(�;�;a)

@� N (�; �; a)

D2 (�; �; a):

It is negative if and only i¤

@N (�; �; a)

@�D (�; �; a) <

@D (�; �; a)

@�N (�; �; a) : (55)

As both @N(�;�;a)@� and @D(�;�;a)

@� are negative, inequality (55) is equivalent to

@N (�; �; a)

@��@D (�; �; a)

@�>N (�; �; a)

D (�; �; a)� T (�; �; a) : (56)

Denote

R (�; �; a) � @N (�; �; a)

@��@D (�; �; a)

@�=

(�4�3a+ 5�+ 9a�+ 7�2

�� 2�

�34a+ 50�+ 62a�+ �2 + 45a2

�+

12�2�a+ 3�+ 3a�+ �2

�+ 16�3 (a+ 1) (a+ 2�))=�

2��14a+ 22�� 118a�� 81�2 � 45a2

�+ 16�3 (a+ 2�� 1) (a+ 2�)

�:

Substituting this notation into inequality (56), we see that T (�; �; a) declines if and only if

R (�; �; a) > T (�; �; a) :

A.6.4 Auxiliary function A (�; �; a), such that R (�; �; a) � A (�; �; a).

De�ne a linear function of � A(a; �; �);

A(a; �; �) = 33a+ �+ 2

9a+ 11�� 2 � 8a+ �+ 8a�+ 6�2 + 2

(9a+ 11�� 2) (37a+ 49�� 6)�:

Let us show that for any � 2 [0; 1]

R (�; �; a) > A(a; �; �):

Consider

�RA = R (�; �; a)�A(a; �; �)

=2 (� � 1)

� (9a+ 11�� 2) (37a+ 49�� 6) �

�[�14a+ 22�� 118a�� 81�2 � 45a2

�+8�2 (a+ 2�� 1) (a+ 2�)]�1

�f�3a+ 5�+ 9a�+ 7�2

�(37a+ 49�� 6) (9a+ 11�� 2) (57)

+� (37a+ 49�� 6) (2a+ 6�+ 176a�+ 105�2

�39�3 � 34a�2 + 9a2�+ 63a2)+8�2 (a+ 2�)

�21a+ 3�� 17a�� 9�2 + 10

�(9a+ 11�� 2)

+32�3 (a+ 2�) (a+ 2�� 1)�a+ �+ 8a�+ 6�2 + 2

�g


Let us look at the signs of components of the product in (57). Note that

2 (� � 1)� ((14a+ 22�� 118a�� 81�2 � 45a2) + 8�2 (a+ 2�� 1) (a+ 2�)) =

4 (� � 1)D0� (�; �; a)

� 0:

The product (9a+ 11�� 2) (37a+ 49�� 6) is positive as each of the factors is positive (aand � are nonnegative and we employ condition (54)):

9a+ 11�� 2 = 8(a+ �) + (a+ 3�� 2) > 0;

37a+ 49�� 6 = 34a+ 40�+ 3 (a+ 3�� 2) > 0:

Similarly, the coe¢ cients of the cubic polynomial with respect to � in the numerator are allpositive: the constant term

�3a+ 5�+ 9a�+ 7�2

�(37a+ 49�� 6) (9a+ 11�� 2) > 0:

The coe¢ cient by � is positive,

(37a+ 49�� 6) (2a+ 6�+ 176a�+ 105�2 � 39�3 � 34a�2 + 9a2�+ 63a2) > 0

as

2a+ 6�+ 176a�+ 105�2 � 39�3 � 34a�2 + 9a2�+ 63a2 �2a+ 6�+ 176a�+ 105�2 � 39�2 � 34a�+ 9a2�+ 63a2 =

a+ 6�+ 142a�+ 66�2 + 9a2�+ 63a2 > 0:

The coe¢ cient by �2 is also positive,

8 (a+ 2�)�21a+ 3�� 17a�� 9�2 + 10

�(9a+ 11�� 2) > 0;

as

21a+ 3�� 17a�� 9�2 + 10 > 21a+ 3�� 17a� 9 + 10 = 4a+ 3�+ 1 > 0:

And, �nally, the coe¢ cient by �3 is positive,

32�3 (a+ 2�) (a+ 2�� 1)�a+ �+ 8a�+ 6�2 + 2

�> 0;

as

a+ 2�� 1 > a+ 3�� 2 � 0:

Thus, we can conclude that for any � 2 [0; 1]

�RA = R (�; �; a)�A(a; �; �) � 0: (58)

A. Appendix 37

A.6.5 Proof of A (�; �; a) � T (�; �; a)

Now we show thatA(a; �; �) � T (�; �; a) :

Consider the di¤erence �TA between T (�; �; a) and A(a; �; �):

�TA = T (�; �; a)�A(a; �; �)

=4�

(9a+ 11�� 2) (37a+ 49�� 6)=[(9a+ 11�� 2) (9a+ 11�)

+�2�14a+ 22�� 118a�� 81�2 � 45a2

�+ 4�4 (a+ 2�� 1) (a+ 2�)]

�((9a+ 11�� 2) (54a+ 74�� 238a�� 181�2

�211�3 � 416a�2 � 189a2�� 93a2) + 2� (37a+ 49�� 6) (59)

��2a� 4�� 29a�� 16�2 + 29�3 + 49a�2 + 18a2�� 9a2

�+�2(68a+ 124�+ 449a2�2 � 912a�� 760�2 + 1555�3 � 433�4

+2585a�2 + 1361a2�� 247a�3 + 279a3�� 280a2 + 243a3)+2�3 (1� �) (37a+ 49�� 6) (5a+ 3�+ 2) (a+ 2�)+8�4 (a+ 2�� 1)

�a+ �+ 8a�+ 6�2 + 2

�(a+ 2�)):

Once more, we determine the signs of the components of the product in (59). We alreadyknow that

(9a+ 11�� 2) (37a+ 49�� 6) > 0:

Moreover,

4�=[(9a+ 11�� 2) (9a+ 11�) + �2�14a+ 22�� 118a�� 81�2 � 45a2

�+ 4�4 (a+ 2�� 1) (a+ 2�)] = 4�

D (�; �; a)� 0:

Consider the remaining component of the product. Denote it by

M (�; �; a) = (9a+ 11�� 2) (54a+ 74�� 238a�� 181�2

� 211�3 � 416a�2 � 189a2�� 93a2) + 2� (37a+ 49�� 6)��2a� 4�� 29a�� 16�2 + 29�3 + 49a�2 + 18a2�� 9a2

�+ �2(68a+ 124�+ 449a2�2 � 912a�� 760�2 + 1555�3 � 433�4

+ 2585a�2 + 1361a2�� 247a�3 + 279a3�� 280a2 + 243a3)+ 2�3 (1� �) (37a+ 49�� 6) (5a+ 3�+ 2) (a+ 2�)+ 8�4 (a+ 2�� 1)

�a+ �+ 8a�+ 6�2 + 2

�(a+ 2�) :

Its second derivative with respect to � is a quadratic parabola,

@2M (�; �; a)

@�2= 2(68a+ 124�+ 449a2�2 � 912a�� 760�2 + 1555�3 � 433�4 +

2585a�2 + 1361a2�� 247a�3 + 279a3�� 280a2 + 243a3) +12� (1� �) (37a+ 49�� 6) (5a+ 3�+ 2) (a+ 2�) +96�2 (a+ 2�� 1)

�a+ �+ 8a�+ 6�2 + 2

�(a+ 2�) :


As (a+ 2�� 1)�a+ �+ 8a�+ 6�2 + 2

�(a+ 2�) > 0, this parabola is convex and has its

minimum at �min = � (1��)(37a+49��6)(5a+3�+2)16(a+2��1)(a+�+8a�+6�2+2) < 0. Thus, it is increasing at � 2 [0; 1]: It

is positive at � = 0,

@2M (0; �; a)

@�2= 2(68a+ 124�+ 449a2�2 � 912a�� 760�2 + 1555�3 � 433�4

+2585a�2 + 1361a2�� 247a�3 + 279a3�� 280a2 + 243a3)= 2[(140a2 + 456a�+ 380�2) (a+ 3�� 2) + 68a+ 124�+ 449a2�2

+415�3 � 433�4 + 837a�2 + 485a2�� 247a�3 + 279a3�+ 103a3]> 2

�124�4 + 415�4 � 433�4 + 837a�2 � 247a�2

�= 4�2

�295a+ 53�2

�> 0;

and increasing in �. So we conclude that for any � 2 [0; 1];

@2M (�; �; a)

@�2> 0:

As a result, M (�; �; a) is a convex function of � at [0; 1] for any (�; a) and reaches itsmaximum at (either of) the corner points of the segment.

But M (�; �; a) is negative both at � = 0 and � = 1. Indeed,

M (0; �; a) = (9a+ 11�� 2) [54a+ 74�� 238a�� 181�2 � 211�3 � 416a�2 � 189a2�� 93a2] < 0;

as

54a+ 74�� 238a�� 181�2 � 211�3 � 416a�2 � 189a2�� 93a2 =(27a+ 37�) (2� 3�� a)�

�120a�+ 70�2 + 211�3 + 416a�2 + 189a2�+ 66a2

�< 0:

Moreover,

M (1; �; a) = �2 (a+ �) (9a+ 11�� 2)�49a+ 81�+ 22a�+ 14�2 � 14

�< 0;

as

49a+ 81�+ 22a�+ 14�2 � 14 = 7(a+ 3�� 2) + 42a+ 60�+ 22a�+ 14�2 > 0:

So we conclude thatM (�; �; a) � 0;

which is equivalent to�TA = T (�; �; a)�A(a; �; �) � 0: (61)

A.6.6 Conclusion: R (�; �; a) � A (�; �; a) � T (�; �; a)

>From (58) and (61), it follows that

R (�; �; a) � T (�; �; a) :

which implies that �1(�;�;a)�01(�)

decreases with � for any admissible (a; �) :

The result that the ratio �2(�;�;a)�02(�)

increases with � for any (a; �) is proven in a similarway.

A. Appendix 39

A.7 Derivation of equation (23)

Equation (23) results from solving system (53) for I1 = I2 = 1; � = 1and �L = 2�. Forthese parameters, the system consists of two symmetric equations�

�4 (1 + a)(a+ 2�)

+ 39

��k �

�4(1 + a)

(a+ 2�)+ 11

��k = (1� c)

�1 + 4

(1 + a)

(a+ 2�)

�;

which yields ee�1 (1; �; a) = ee�2 (1; �; a) = (1� c)4

5a+ 2�+ 4

5a+ 14�� 2 :

A.8 Derivation of equations (34) and (35)

>From equations (31) and (33), it follows that

V1(ee� )� V1(e� ) = W1(ee� )� 12

�aW (� 0)� aW (ee� )�� W1(e� )� aW (� 0) + aW (e� )�

= W1(ee� )�W1(e� ) + 12a�W (� 0) +W (ee� )� 2W (e� )� : (62)

Substituting the expressions for the outputs, pro�ts, volume of imports and domestic con-sumption into the formulas for the lobby i = 1; 2, welfare (6) and aggregate social welfare(7), and simplifying the resulting expressions yields

Wi(ee� ) = 9

4(1� c)2 (a+ 2�) a� 2�+ 14�

2 + 3a�

(5a+ 14�� 2)2;

Wi(e� ) = 9

4(1� c)2 (a+ �) a� �+ 7�

2 + 3a�

(5a+ 7�� 1)2;

W (� 0) =9

20(1� c)2 ;

W (e� ) =94(1� c)2 (a+ �) 5a+ 9�� 2

(5a+ 7�� 1)2;

W (ee� ) =94(1� c)2 (a+ 2�) 5a+ 18�� 4

(5a+ 14�� 2)2:

Inserting into (62), we obtain

V1(ee� )� V1(e� ) = 9

20

(1� c)2 a (1� 2�)2

(5a+ 7�� 1) (5a+ 14�� 2) :

Similarly, equations (32), (33) and the formulas above imply that

V2(ee� )� V2(e� ) = W1(ee� )� 12

�aW (� 0)� aW (ee� )��W2(e� )

=9

20

(1� c)2 a (1� 7�) (1� 2�)2

(5a+ 14�� 2) (5a+ 7�� 1)2:


A.9 Proof of Lemma (8)

If industry 2 chooses ffB2 de�ned by expression (33), industry 1�s best response is preciselysetting ffB1 = ffB2. Clearly, it is not in industry 1�s interest to decrease ffB1 (and hence itspayo¤, in case ffB1 does not bind). We need to consider two cases. Assume �rst that � > 1=7so that ffBi =Wi(ee� )� 1

2

�aW (� 0)� aW (ee� )� < W1(e� );

which implies that the government gets positive contributions both at policies ee� and e� . Ifindustry 1 chooses ffB1 > ffB2, the government prefers tari¤ � 0 to ee� and e� . Indeed, by equation(8) ee� maximizes the sum of social welfare and the lobbies�welfare, so that the governmentprefers ee� to e� :

G (e� ) = aW (e� )+�W1(e� )� ffB1�+ �W2(e� )� ffB2�< aW (ee� )+�W1(ee� )� ffB1�+ �W2(ee� )� ffB2� = G(ee� ):

In turn, due to our assumption ffB1 > ffB2; policy ee� does not pay the government enough todeviate from the �rst-best policy:

G(ee� ) = aW (ee� )+�W1(ee� )� ffB1�+ 12

�aW (� 0)� aW (ee� )�

=

�W1(ee� )� ffB1�+ 1

2

�aW (� 0) + aW (ee� )� < aW (� 0):

If instead � � 1=7, implying that

ffBi =Wi(ee� )� 12

�aW (� 0)� aW (ee� )� > W1(e� );

the government never sets e� as it does not get any contributions for this policy. Similarly toabove, the government prefers tari¤ � 0 to ee� . Therefore, by increasing ffB1 industry 1 receivesa payo¤W1(� 0) <

ffB1, so this cannot be a best response.

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